Weyl's tube formula

Weyl's tube formula gives the volume of an object defined as the set of all points within a small distance of a manifold.

Let $\Sigma$ be an oriented, closed, two-dimensional surface, and let $N_{\varepsilon }(\Sigma )$ denote the set of all points within a distance $\varepsilon$ of the surface $\Sigma$ . Then, for $\varepsilon$ sufficiently small, the volume of $N_{\varepsilon }(\Sigma )$ is

V=2A(\Sigma )\varepsilon +{\frac {4\pi }{3}}\chi (\Sigma )\varepsilon ^{3},

where $A(\Sigma )$ is the area of the surface and $\chi (\Sigma )$ is its Euler characteristic. This expression can be generalized to the case where $\Sigma$ is a $q$ -dimensional submanifold of $n$ -dimensional Euclidean space $\mathbb {R} ^{n}$ .